The minimum-codegree conjecture for squares of tight Hamilton cycles in 3-graphs

Let $H$ be a $3$-graph on $n$ vertices, and let $\delta_2(H)$ denote its minimum codegree, the minimum number of vertices completing a pair to an edge of $H$. The square of a tight Hamilton cycle is the spanning $3$-graph obtained by taking every triple contained in four consecutive vertices of a cyclic ordering. Minimum-codegree conjecture. For every $\alpha>0$ there exists $n_0\in\mathbb{N}$ such that every $3$-graph $H$ on $n\geq n_0$ vertices with

$$\delta_2(H) \geq (3/4+\alpha)n$$

contains the square of a tight Hamilton cycle. This conjecture improves the sufficient bound proved in the paper from $7n/9+o(n)$ toward the expected threshold $3n/4$. The source describes it as a highly significant open problem in extremal graph theory.